A note on the paper “Optimizing improved Hardy inequalities” by S. Filippas and A. Tertikas (original) (raw)

Cocompactness and minimizers for inequalities of Hardy–Sobolev type involving N-Laplacian

Nonlinear Differential Equations and Applications NoDEA, 2010

The paper studies quasilinear elliptic problems in the Sobolev spaces W 1,p (Ω), Ω ⊂ R N , with p = N , that is, the case of Pohozhaev-Trudinger-Moser inequality. Similarly to the case p < N where the loss of compactness in W 1,p (R N) occurs due to dilation operators u → t (N −p)/p u(tx), t > 0, and can be accounted for in decompositions of the type of Struwe's "global compactness" and its later refinements, this paper presents a previously unknown group of isometric operators that leads to loss of compactness in W 1,N 0 over a ball in R N. We give a one-parameter scale of Hardy-Sobolev functionals, a "p = N "-counterpart of the Hölder interpolation scale, for p > N, between the Hardy functional |u| p |x| p dx and the Sobolev functional |u| pN/(N −mp) dx. Like in the case p < N, these functionals are invariant with respect to the dilation operators above, and the respective concentration-compactness argument yields existence of minimizers for W 1,N-norms under Hardy-Sobolev constraints.

A note on Hardy's inequalities

2016

Let Ω be a smooth bounded domain in R N with N ≥ 1. In this paper we study the Hardy-Poincaré inequalities with weight function singular at the boundary of Ω. In particular we give sufficient conditions so that the best constant is achieved.

The Hardy–Morrey & Hardy–John–Nirenberg inequalities involving distance to the boundary

Journal of Differential Equations, 2016

We strengthen the classical inequality of C. B. Morrey concerning the optimal Hölder continuity of functions in W 1,p when p > n, by replacing the L p-modulus of the gradient with the sharp Hardy difference involving distance to the boundary. When p = n we do the same strengthening in the integral form of a well known inequality due to F. John and L. Nirenberg.

Sharp Hardy–Sobolev inequalities

Comptes Rendus Mathematique, 2004

Let Ω be a smooth bounded domain in IR N , N ≥ 3. We show that Hardy's inequality involving the distance to the boundary, with best constant (1/4), may still be improved by adding a multiple of the critical Sobolev norm.

Critical Hardy–Sobolev inequalities

Journal de Mathématiques Pures et Appliquées, 2007

We consider Hardy inequalities in R n , n 3, with best constant that involve either distance to the boundary or distance to a surface of co-dimension k < n, and we show that they can still be improved by adding a multiple of a whole range of critical norms that at the extreme case become precisely the critical Sobolev norm.

Reversed Hardy-Littewood-Sobolev inequality

2014

The classical sharp Hardy-Littlewood-Sobolev inequality states that, for 1<p, t<∞ and 0<λ=n-α <n with 1/p +1 /t+ λ /n=2, there is a best constant N(n,λ,p)>0, such that |∫_R^n∫_R^n f(x)|x-y|^-λ g(y) dx dy|< N(n,λ,p)||f||_L^p(R^n)||g||_L^t(R^n) holds for all f∈ L^p(R^n), g∈ L^t(R^n). The sharp form is due to Lieb, who proved the existence of the extremal functions to the inequality with sharp constant, and computed the best constant in the case of p=t (or one of them is 2). Except that the case for p∈ ((n-1)/n, n/α) (thus α may be greater than n) was considered by Stein and Weiss in 1960, there is no other result for α>n. In this paper, we prove that the reversed Hardy-Littlewood-Sobolev inequality for 0<p, t<1, λ<0 holds for all nonnegative f∈ L^p(R^n), g∈ L^t(R^n). For p=t, the existence of extremal functions is proved, all extremal functions are classified via the method of moving sphere, and the best constant is computed.

On a higher-order Hardy inequality

Mathematica Bohemica

The Hardy inequality ∫ Ω |u(x)| p d(x) -p dx≤c∫ Ω |∇u(x)| p dx with d(x)=dist(x,∂Ω) holds for u∈C 0 ∞ (Ω) if Ω⊂ℝ n is an open set with a sufficiently smooth boundary and if 1<p<∞. P. Hajłasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for p=1.

Sharp Hardy–Littlewood–Sobolev Inequality on the Upper Half Space

International Mathematics Research Notices, 2013

There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent lambda=n−alpha\lambda=n-\alphalambda=n−alpha (that is for the case of alpha>n\alpha>nalpha>n). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via the method of moving sphere.